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| -rw-r--r-- | stokes.tex | 91 |
1 files changed, 50 insertions, 41 deletions
@@ -911,7 +911,7 @@ complex matrix $B$. Introducing replicas to bring the partition function into the numerator of the Green function \cite{Livan_2018_Introduction} gives -\begin{equation} \label{eq:green.replicas} +\begin{equation} \label{eq:green.replicas} \fl\quad G(\sigma)=\lim_{n\to0}\int dz\,dz^*\,(z^{(0)})^\dagger z^{(0)} \exp\left\{ -\sum_\alpha^n\left[(z^{(\alpha)})^\dagger z^{(\alpha)}\sigma @@ -927,11 +927,13 @@ replica vectors. The replica-symmetric ansatz leaves all replica vectors zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$, $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is \begin{equation}\label{eq:green.saddle} - \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0 - \exp\left\{nN\left[ - 1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2) - +\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right) - \right]\right\}. + \eqalign{ + \overline G(\sigma)=N\lim_{n\to0}\int &d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0 + \exp\left\{nN\left[ + 1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2\right.\right.\cr + &\left.\left.+\frac12\log(\alpha_0^2-|\chi_0|^2)+\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right) + \right]\right\}. + } \end{equation} \begin{figure} @@ -1256,16 +1258,18 @@ the result can be written, neglecting constant factors, \overline\mathcal N\simeq\int dQ\,e^{NS_\mathrm{eff}(Q)} \end{equation} for an effective action functional of the supermatrix $Q$ -\begin{equation} - S_{\mathrm{eff}}= - \int d1\,d2\,\operatorname{Tr}\left( - \frac14\left[ - \matrix{\frac14&\frac14\cr\frac14&\frac14} - \right]Q^{(p)}(1,2)-\frac p2\left[ - \matrix{\frac\epsilon2&0\cr0&\frac{\epsilon^*}2} - \right](Q(1,1)-I)\delta(1,2) - \right) - +\frac12\log\det Q +\begin{equation} \fl + \eqalign{ + S_{\mathrm{eff}}&= + \int d1\,d2\,\operatorname{Tr}\left( + \frac14\left[ + \matrix{\frac14&\frac14\cr\frac14&\frac14} + \right]Q^{(p)}(1,2)-\frac p2\left[ + \matrix{\frac\epsilon2&0\cr0&\frac{\epsilon^*}2} + \right](Q(1,1)-I)\delta(1,2) + \right) \\ + &\hspace{28em}+\frac12\log\det Q + } \end{equation} where the exponent in parentheses denotes element-wise exponentiation, and \begin{equation} @@ -1382,23 +1386,28 @@ stationary points with given energies $\epsilon_1$ and $\epsilon_2$ are \right] \end{equation} \begin{equation} - S_\mathrm{eff} - =\int d1\,d2\,\operatorname{Tr}\left\{ - \frac14\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]Q^{(p)}(1,2) - -\frac p2\left[ - \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*} - \right]\left( - Q(1,1)-I - \right)\delta(1,2) - \right\}+\frac12\det Q + \eqalign{ + S_\mathrm{eff} + &=\int d1\,d2\,\operatorname{Tr}\left\{ + \frac14\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]Q^{(p)}(1,2) + \right. \\ + &\qquad\qquad\left.-\frac p2\left[ + \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*} + \right]\left( + Q(1,1)-I + \right)\delta(1,2) + \right\}+\frac12\det Q + } \end{equation} \begin{equation} - 0=\frac{\partial S_\mathrm{eff}}{\partial Q(1,2)} - = - \frac p4\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,2) - -\frac p2\left[ - \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}\right]\delta(1,2) - +\frac12Q^{-1}(1,2) + \eqalign{ + 0=\frac{\partial S_\mathrm{eff}}{\partial Q(1,2)} + &= + \frac p4\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,2) \\ + &\qquad\qquad-\frac p2\left[ + \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}\right]\delta(1,2) + +\frac12Q^{-1}(1,2) + } \end{equation} where $\odot$ denotes element-wise multiplication. \begin{equation} @@ -1406,8 +1415,8 @@ where $\odot$ denotes element-wise multiplication. 0 &=\int d3\,\frac{\partial S_\mathrm{eff}}{\partial Q(1,3)}Q(3,2) \\ &=\frac p4\int d3\, - \left\{\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,3)\right\}Q(3,2) - -\frac p2 \left[ + \left\{\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,3)\right\}Q(3,2) \\ + &\qquad\qquad\qquad\qquad -\frac p2 \left[ \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}\right]Q(1,2) +\frac12I\delta(1,2) } @@ -1478,7 +1487,7 @@ function of $\Delta$ and $\arg\delta$. } \end{figure} -\subsection{Pure $p$-spin: is analytic continuation possible?} +\subsection{Pure {\it p}-spin: is analytic continuation possible?} \begin{equation} \eqalign{ @@ -1489,7 +1498,7 @@ function of $\Delta$ and $\arg\delta$. e^{-\beta\mathcal S(s_\sigma)} \\ &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k) \left(\frac{2\pi}\beta\right)^{D/2}i^k - \left(|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}\bigm|\mathcal S(s_\sigma)=N\epsilon,k_\sigma=k\right) e^{-\beta N\epsilon} + |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} e^{-\beta N\epsilon} } \end{equation} Following Derrida \cite{Derrida_1991_The}, @@ -1503,17 +1512,17 @@ governs things at large $|\beta|$, not its total. This gives two terms to the ty \begin{equation} Z_\mathrm{typ}=Z_A+Z_B \end{equation} -\begin{eqnarray} +\begin{eqnarray} \fl Z_A - &\simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k) + \simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k) \left(\frac{2\pi}\beta\right)^{D/2}i^k |\det\operatorname{Hess}\mathcal S|^{-\frac12} e^{-\beta N\epsilon} =\int d\epsilon\,e^{Nf_A(\epsilon)} - \\ + \\ \fl Z_B - &\simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2} + \simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2} \left(\frac{2\pi}\beta\right)^{D/2}i^k - |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} + |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} \\ =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)} \end{eqnarray} for @@ -1577,7 +1586,7 @@ large-$|\beta|$ saddle-point used to evaluate the thimble integrals. Taking the thimbles to the next order in $\beta$ may reveal more explicitly where Stokes points become important. -\section{The $p$-spin spherical models: numerics} +\section{The {\it p}-spin spherical models: numerics} To study Stokes lines numerically, we approximated them by parametric curves. If $z_0$ and $z_1$ are two stationary points of the action with |